Berkeley Program in Law and Economics UC Berkeley

Transcription

1 Berkeley Program in Law and Economics UC Berkeley Title: Per-Mile Premiums for Auto Insurance Author: Edlin, Aaron S., School of Law and Economics Dept., University of California, Berkeley Publication Date: Series: Berkeley Program in Law and Economics, Working Paper Series Permalink: Abstract: Most insurance premiums are only weakly linked to mileage, and have largely lump-sum characteristics. The probable result is too many accidents and too much driving from the standpoint of economic efficiency. This paper develops a model of the relationship between driving and accidents that formalizes Vickrey's [1968] central insights about the accident externalities of driving. We use it to estimate the driving, accident, and congestion reduc- tions that could be expected from switching to other insurance pricing systems. Under a competitive system of permile premiums, in which insurance companies quote risk-classified per-mile rates, we estimate that the reduction in insured accident costs net of lost driving benefits would be $9.8 -$12.7 billion in the U.S., or $58-$75 per insured vehicle. When congestion reductions are considered, the net benefits rise to $15-$18 billion, exclusive of monitoring costs. The total benefits of per-mile premiums with a Pigouvian tax to account for accident externalities would be $19-$25 billion, or $111-$146 per insured vehicle, ex- clusive of monitoring costs. Accident externalities may go a long way toward explaining why most insurance companies have not switched to per-mile premiums despite these large potential social benefits. Copyright Information: All rights reserved unless otherwise indicated. Contact the author or original publisher for any necessary permissions. escholarship is not the copyright owner for deposited works. Learn more at escholarship provides open access, scholarly publishing services to the University of California and delivers a dynamic research platform to scholars worldwide.

2 1 Per-Mile Premiums for Auto Insurance Aaron S. Edlin UC Berkeley and NBER Draft Printed: June 18, 2002 Mailing address: 549 Evans Hall Department of Economics UC Berkeley Berkeley, California ************************************************************* Composed using speech recognition software. Misrecognized words are common. Imagination is sometimes helpful... ************************************************************* I am grateful for a Faculty Fellowship from the Alfred P. Sloan Foundation, support from the World Bank, a grant from the UC Berkeley Committee on Research, Visiting Olin Fellowships at Columbia Law School and Georgetown University Law Center, and for the comments and assistance of George Akerlof, Richard Arnott, Severin Borenstein, Patrick Butler, Amy Finkelstein, Steve Goldman, Louis Kaplow, Todd Litman, Eric Nordman, Mark Rainey, Zmarik Shalizi, Joseph Stiglitz, Steve Sugarman, Jeroen Swinkels, Michael Whinston, Janet Yellen, Lan Zhao, several helpful people in the insurance industry, and seminar participants at Cornell, Georgetown, New York University, the University of Toronto, the University of Pennsylvania, the University of Maryland, The National Bureau of Economic Research, and The American Law and Economics Association Annual Meetings. The opinions in this paper are not necessarily those of any organization with whom I have been affiliated.

3 2 Per-Mile Premiums for Auto Insurance ABSTRACT Most insurance premiums are only weakly linked to mileage, and have largely lump-sum characteristics. The probable result is too many accidents and too much driving from the standpoint of economic efficiency. This paper develops a model of the relationship between driving and accidents that formalizes Vickrey s [1968] central insights about the accident externalities of driving. We use it to estimate the driving, accident, and congestion reductions that could be expected from switching to other insurance pricing systems. Under a competitive system of per-mile premiums, in which insurance companies quote risk-classified per-mile rates, we estimate that the reduction in insured accident costs net of lost driving benefits would be $9.8 -$12.7 billion in the U.S., or $58-$75 per insured vehicle. When congestion reductions are considered, the net benefits rise to $15-$18 billion, exclusive of monitoring costs. The total benefits of per-mile premiums with a Pigouvian tax to account for accident externalities would be $19-$25 billion, or $111-$146 per insured vehicle, exclusive of monitoring costs. Accident externalities may go a long way toward explaining why most insurance companies have not switched to per-mile premiums despite these large potential social benefits.

4 1...the manner in which [auto insurance] premiums are computed and paid fails miserably to bring home to the automobile user the costs he imposes in a manner that will appropriately influence his decisions. William Vickrey Americans drive 2,360,000,000,000 miles each year, and the cost of auto accidents is commensurately large: 1 roughly $100 billion in accident insurance, 2 andaccordingtothe Urban Institute [1991] an additional $250 billion in uninsured accident costs per year. Every time a driver takes to the road, and with each mile she drives, she exposes herself and others to the risk of accident. (The degree of risk depends, of course, upon a wide variety of factors such as a driver s skill or age, and the territory she drives in.) Yet, most auto insurance premiums have largely lump-sum characteristics and are only weakly linked to mileage. Mileage classifications are coarse, and low-mileage discounts are extremely modest and based on self-reported estimates of future mileage that have no implicit or explicit commitment. 3 (Two noteworthy exceptions are premiums on some commercial policies 4 and a few recent pilot programs.) 5 Few drivers therefore pay or perceive a significant 1 See table No. 1030, Statistical Abstract of the United States, 1997, U.S. Department of Commerce. Figure for After subtracting comprehensive insurance coverage, which covers fire, theft, vandalism and other incidents unrelated to the amount of driving, the remaining premiums for private passenger vehicles totaled $84 billion in State Average Expenditures and Premiums for Personal Automobile Insurance in 1995, National Association of Insurance Commissioners, Jan In additon, commercial premiums are approximately 15 percent of premiums for private passenger vehicles. The Insurance Information Institute 1998 Fact Book, p For example, State Farm distinguishes drivers based upon whether they report an estimated annual mileage of under or over 7500 miles. Drivers who estimate annual mileages of under 7500 miles receive 15% discounts (5% in Massachusetts). The 15% discount is modest given that those who drive less than 7500 miles per year drive an average of 3600 miles compared to 13,000 miles for those who drive over 7500 per year, according to the 1994 Residential Transportation Energy Consumption Survey of the Department of Energy. The implied elasticity of accident costs with respect to miles is.05, anorderofmagnitudebelow what the evidence suggests is the private or social elasticity of accident costs. The link between driving and premiums may be attenuated in part because there is significant noise in self-reported estimates of future mileage, estimates whose accuracy does not affect insurance pay-outs. Insurance companies also classify based upon the distance of a commute to work. These categories are also coarse, however. State Farm, for example, classifies cars based upon whether they are used for commuting less than 20 miles per week, in between 20 and 100 miles per week, or over 100 miles per week. 4 For private and public livery, taxicabs, and buses, because rates are high and because there is no risk when the car is not in operation, a system of rating has been devised on an earnings basis per $100 of gross receipts or on a mileage basis. Bickelhaupt [1983, p. 613]. For details on per-mile commercial insurance, see Commercial Automobile Supplementary Rating Procedures, Insurance Services Office, on file with author. 5 OneexperimentisinTexasandanotherintheUK.See InsurancebytheMile, WallStreetJour-

5 2 insurance cost from driving an extra mile, despite the substantial accident costs involved. An ideal tort and insurance system would charge each driver the full social cost of her particular risk exposure on the marginal mile of driving. Otherwise, people will drive too much and cause too many accidents (from the vantage of economic efficiency). In principle, insurance companies could levy a substantial charge for driving an extra mile, as new car leases do; however, this would require them to incur the cost of verifying mileage (through periodic odometer checks or by installing a monitoring and broadcasting device in vehicles). 6 A central point of this paper is that externalities make their incentives to do so considerably less than the social incentives. If insurance company C is able to reduce the driving of its insureds, although it will save on accident payouts, substantial external savings will be realized by other insurance carriers and their insureds who will get into fewer accidents with C s insureds. These externalities follow from Vickrey s observation that if two drivers get into an accident, even the safer driver is typically a but for cause of the accident in the sense that had she opted for the metro, the accident would not have occurred. 7 Externalities help explain why we are only just now seeing pilot per-mile premiums programs. Accident externalities suggest a valuable role for policy, and this paper investigates the nal (1999), business/newsid / stm, Progressive Corporation has a pilot program in Texas in which miles (and locations) are monitored by a device using cellular-phone and satellite technology. Since January 2002 it is now legal for auto insurance policies in Texas to have the unit of exposure be the vehicle-mile rather than the more traditional vehicle-year. 6 In practice there may be a regulatory constraint that has discouraged firms from per-mile pricing. The traditional unit of risk insurance that insurance companies price is the vehicle-year. Just recently after a lobbying campaign by the National Organization for Women, Texas adopted legislation allowing firms to adopt the vehicle-mile as the unit of exposure. ( Long before this, however, firms charged slight surcharges based upon unaudited reports of high usage. Obviously, there is a fuzzy line here because as such charges become more refined and audits of usage are performed, the vehicleyear exposure unit slips into a vehicle-mile unit. Firms have not generally pushed that direction, however, so it is unclear whether there has been a meaningful regulatory constraint. 7 Sometimes, of course, only one driver is the cause of an accident, even when the accident involves multiple cars such as when a driver plows into a long line of cars. If one car wasn t there to absorb the impact, another would have, so the cars that are hit do not cause the accident in any respect. Such accident substitution is not accounted for by the theoretical model we present, and reduces the externalities from driving. This substitution effect is, however, accounted for by our regression results.

6 3 potential benefits of two proposals that would increase the marginal charge for driving, and consequently reduce driving and accidents. The first proposal is per-mile premiums, advocated by Litman [1997], Butler [1990], and the National Organization for Women [1998]. Under a per-mile premium system, the basic unit of exposure would shift from the caryear to the car-mile, either by requirement or by subtler policy tools, so that the total premiums of driver i would be m i p i,wherem i is the miles i travels and p i is the per-mile rate. An individual s per-mile rate, p i, would vary among drivers to reflect the per-mile risk of a given driver and could depend upon territory, driver age, safety records or other relevant characteristics used today for per-year rates. (In fact, the technology now used experimentally by Progressive in Texas also allows prices to vary by time of the day and by location.) 8 The second proposal is to couple per-mile premiums with a Pigouvian tax in order to account for the Vickrey accident externality. Both these proposals differ fundamentally from the uniform per-gallon gas tax proposals of Vickrey [1968], 9 Sugarman [1993], and Tobias [1993], because under gas tax proposals, unlike per-mile premiums, the additional cost of driving would be independent of driver age, driver safety records, or in some cases of territory (all highly important indicia of risk), yet would depend upon fuel efficiency ( a relatively poor risk measure). This paper makes several contributions. We begin by developing a simple model from primitives that relates miles driven to accidents, formalizing Vickrey s insights about the externalities of driving this contribution is mainly pedagogical. Our second contribution is to provide evidence that these externalities are substantial. Our third is to provide the first estimates of the potential benefits of per-mile premiums that take into account Vickrey s externalities as well as the resulting fact that as driving falls, so too will accident rates and 8 See Wall Street Journal [1999]. 9 Actually, Vickrey s first suggestion was that auto insurance be bundled with tires hoping that the wear on a tire would be roughly proportional to the amount it is driven. He worried about moral hazard (using a tire until it was threadbare), but concluded that this problem would be limited if refunds were issued in proportion to the amount of tread remaining.

7 4 per-mile premiums. 10 Our fourth contribution is to estimate the benefits of a per-mile premium policy coupled with a Pigouvian tax. (It s natural to consider taxing per-mile premiums to account for accident externalities once one incorporates externalities.) Finally our estimates incorporate lower bound estimates of congestion cost reductions. These estimates are a rough first cut and should be viewed as lower bounds for reasons we will elaborate. Our evidence that accident externalities are significant in practice is that states with more traffic density have considerably higher insurance costs per mile driven. This suggests that the more people drive on the same roads, the more dangerous driving becomes. (A little introspection will probably convince most readers that crowded roadways are more dangerous than open ones. In heavy traffic, most us feel compelled to a constant vigilance to avoid the numerous moving hazards.) 11 Nationally, the insured cost of accidents is roughly 4 cents per-mile driven, but we estimate that the marginal cost the cost if an extra mile is driven is much higher, roughly 7 and a half cents, because of these accident externalities. In high traffic-density states like New Jersey, Hawaii, or Rhode Island, we estimate that the marginal cost is roughly 15 cents. For comparison, gasoline costs roughly 6 cents per mile, so an efficient Pigouvian charge for accidents at the margin would dramatically increase the marginal cost of driving, and would presumably reduce driving substantially. Even without a Pigouvian charge to account for accident externalities, a system of per-mile premiums that shifted a fixed insurance charge to the margin would be roughly equivalent to a 70% hike in the gasoline price and could be expected to reduce driving 10 Externalities turn out to increase the benefit estimates by %, over what one would calculate in a linear model of accidents (i.e. a model without externalities) as studied by Litman [1987)] and Rea [1992]. Note that a linear model is appropriate to estimate the gains to a given insurance carrier with small market shareanditscustomersfromswitchingtoper-milepremiumsasreadoes. 11 Thisvigilancenodoubtworkstooffset the dangers we perceive but seems unlikely to completely counter balance them. Note also that the cost of stress and tension that we experience in traffic are partly accident avoidance costs and should properly be included in a full measure of accident externality costs.

8 5 nationally by 9.2% - 9.5%, and insured accident costs by $14 - $17 billion. After subtracting thelostdrivingbenefits of $4.3 - $4.4 billion, the net accident reductions would be $9.8 - $12.7 billion or $58 - $75 per insured vehicle. The net savings would be $ $15.3 billion if per-mile premiums were taxed to account for the external effect of one person s driving on raising others insurance premiums. 12 These estimates are probably a lower bound on what savings would actually be under a per-mile system. The reason is that these estimates use state level data and assume that drivers and territories are homogeneous within a state. Currently intrastate heterogeneity inaccident risksandcostsisreflected in yearly insurance premiums. In a per-mile system, this heterogeneity would likewise be reflected in per-mile rates that vary substantially by territory, driver age, and driver accident record. Since the most dangerous drivers in the most dangerous territories would face the steepest rise in marginal driving cost and therefore reduce driving the most, actual benefits could be considerably larger than our estimates. If state heterogeneity is a useful guide, territory heterogeneity alone would raise the benefits of per-mile premiums by 10%. Other types of heterogeneity (such as age) could raise benefits substantially more. The main reason insurance companies have not switched to per-mile premiums is probably that monitoring actual mileage with yearly odometer checks seems too costly given their potential gains, as suggested by Rea [1992] and Williamson et al. [1967, p. 247]. 13 However, our analysis suggests that the gains a given insurance company could realize by switching to per-mile premiums are considerably less than the social gains. A single company and its customers might stand to gain only $31 per vehicle per year from the switch, far less than the potential social gains of $58 per insured vehicle that we estimate when we 12 These ranges represent point estimates obtained with the regression and calibration methods described in section Monitoring costs are cited as the principal reason by actuaries I have interviewed (see also Nelson [1990] and Cardoso [1993]).

9 6 include the Vickrey externality (i.e., the reduction in others insurance costs.) Moreover, the $31 in private gains would be temporary from an insurer s vantage, and would all go to consumers once other firms match its new policies. 14 This discrepancy implies that the social gains from per-mile premiums might justify the monitoring costs (and the fixed costs of transition), even if no single insurance company could profit from the change itself. Other external benefits could make the discrepancy between the private gains from permile premiums and the social gains even larger. A great deal of accident costs are uninsured or underinsured (more than half according to the Urban Institute) and the driving reductions caused by per-mile premiums should reduce these costs just as they reduce insured accident costs. 15 Policy intervention looks more attractive still when nonaccident benefits such as congestion are taken into account. Congestion reductions raise our estimates of the benefits from per-mile premiums by $5.5-$5.7 billion. This brings our estimates of total national benefits from per-mile premiums to $15.5-$18.2 billion ($ billion with a Pigouvian tax), or $ per insured vehicle ($ with a Pigouvian tax). Benefits would be higher still, if pollution costs, road maintenance costs, and other externality costs are higher than we assume here. 16 Thefactthataccidentandcongestion externalities could make up more than two thirds of the benefits from per-mile premiums suggests that even if monitoring costs are so large that it is rational for insurance companies to maintain the current premium structure, it is likely that per-mile premiums could still enhance efficiency in many states. Likewise, it suggests that as mileage monitoring technology becomes cheaper (e.g., cellular phone and global positioning system technology), insurance companies may be slower at adopting these technologies than is socially efficient. 14 In a competitive industry, insurance companies cannot profit from a coordinated change, because the efficiency gains would be competed away in lower prices. 15 See e.g. Dewees, Duff and Trebilcock [1996] for evidence of substantial undercompensation. See also the estimates of the Urban Institute [1991]. 16 We assume in this article that existing gasoline taxes of cents per gallon account for these costs. Many estimates, however, suggest that these costs may be much higher. Delucci [1997] estimates that the pollution costs of motor vehicles in terms of extra mortality and morbidity are $26.5-$461.9 billion per year in the U.S.

10 7 Some caution is required, of course, in relying upon this paper s estimates. This paper offers only a first-cut estimate of the accident externality effect, and the benefit estimates rely upon highly uncertain estimates of the price sensitivity of driving. Nonetheless, the estimates are large enough to strongly recommend further research, and provide some support for policy reform. Section 1 presents a simple model of accidents that formalizes Vickrey s insights about accident externalities and incorporates congestion. Section 2 describes the data. Section 3 estimates the marginal accident cost of driving. Section 4 simulates driving and accident reductions under per-mile premiums. Section 5 concludes and explores the policy implications of this research. 1 A Simple Model of Accidents and Congestion. We now develop a model relating driving to accidents and use it to simulate the consequences of various pricing scenarios. For simplicity, we construct an entirely symmetric model in which drivers, territory, and roads are undifferentiated and identical. The central insights continue to hold in a world where some drivers, roads, and territories are more dangerous than others, with some provisos. The relationship between aggregate accidents and aggregate miles will only hold exactly if the demand elasticity is the same across types of driving and drivers. Otherwise, accidents will be either more or less responsive to driving according to whether extra miles are driven by more or less dangerous drivers under more or less dangerous conditions. We also limit attention to one and two vehicle accidents, ignoring the fact that many accidents only occur because of the coincidence of three or more cars. 17 We treat accidents involving two or more cars as if they all involve only two cars because multi-vehicle accidents are not separated in our accident data. Refined data would increase our estimates of the 17 For example, one car may stop suddenly causing the car behind to switch lanes to avoid a collision the accident occurs only if another car is unluckily in the adjacent lane.

11 8 benefits from the driving reductions associated with per-mile premiums because the size of accident externalities increase with the number of cars involved in collisions. Let m i = miles traveled by driver i per year M = aggregate vehicle miles traveled per year by all drivers l =totallanemiles D =traffic density,ortraffic volume=m/ l f i = probability that i is driving at any given time δ 1 = damages from one-vehicle accident δ 2 = damages to each car in a two-vehicle accident Holding speed constant, the fraction of the time that i is driving, f i, will be proportional tothemilesshedrives,m i ; hence f i = ρm i,forsomeρ. For convenience, imagine that the l lane miles are divided into L locations of equal length. An accident occurs between driver i and j if they are in the same location and neither brakes or takes other successful evasive action. The chance that i is driving and j isinthesamelocationisf i (f j /L). Let q be the probability of accident conditional upon being in the same location. The expected rate of damages to i from two-car accidents with j 6= i will then be a 2i,j = δ 2 f i f j L q. Summing over j 6= i and substituting ρm j for f j and ρm i for f j yields expected damages to i from two-car accidents: Letting c 2 δ 2 ρ 2 l/l, wehave a 2i = δ 2 ρ 2 qσ j6=i m j m i. L (M m i ) a 2i = c 2 m i, l

12 9 or, assuming m i is small relative to M, a 2i c 2 m i M l = c 2 m i D. Ignoring multiple-car accidents, the total expected accident damages suffered by driver i are then a i = c 1 m i + c 2 m i D. The first term in the equation reflects the fact that a driver may be involved in an accident even if he is driving alone (e.g., fallingasleepatnightanddrivingintoatree), with c 1 representing the expected accident costs from driving a mile alone. The second term reflects the fact that the chance of getting into an accident with other vehicles in that mile increases as the traffic densityd increases. The linearity of this model in m i ignores the possibility that practice and experience could bring down the per-mile risk, as well as the offsetting possibility that driving experience (which is generally a safe experience) could lead to complacency and conceit. Empirical estimates of the elasticity of an individual s accidents with respect to that individual s mileage, as surveyed in Edlin [1999], range from.35 to.92, but as Edlin [1999] discusses, this work has been limited by the scarcity of reliable microlevel data pairing mileage and accidents, and probably yields downward biased estimates because of noisy mileage data and the difficulty of controlling for the factors that cause any given driver to drive very little (which are likely related to accident propensity). 18 Summing over each driver i yields the total accident costs: A = c 1 M + c 2 MD = c 1 M + c 2 M 2 /l. (1) Observe that the cost of two-car accidents c 2 M 2 /l increases with the square of total 18 For an example of such a downward bias, consider Hu et al. [1998] who study an elderly population. Omitted bad health variables seem likely to be positively correlated with worse driving and probably with less driving as well. Mileage data in that study also come from survey and seem highly susceptible to measurement error.

13 10 miles. Aggregate accident costs are quadratic in aggregate vehicle miles traveled, and this non-linearity is the source of the externality effect. The marginal total accident cost from driving an extra mile is da dm = c 1 +2c 2 D. (2) In contrast, the marginal cost of accidents to driver i is only da i dm i = c 1 + c 2 D. (3) The difference between these two costs, c 2 D, is the externality effect. It represents the fact that when driver i gets in an accident with another driver he is typically the but for cause of both drivers damages in the sense that, but for him having been driving, the accident would not have happened. (Strangely enough, it is entirely possible that both drivers are the but for cause of all damages). This model could overstate the externality effect because of accident substitution: i.e., because if driver A and B collide, it is possible that driver A would have hit driver C if driver B weren t there. 19 Such a substitution effect would be captured in our regression estimates by a lower coefficient on traffic density,and hence a lower estimate of the externality effect. A different view of the accident externality of driving is found by observing that the average cost of accidents per mile driven is: A M = c 1 + c 2 D. (4) A given driver who drives the typical mile expects to experience the average damages A M. Yet, this driver also increases D, which means that he also causes the accident rate for 19 On the other hand, it understates the externality effect to the extent that some collisions require more than two vehicles.

14 11 others to rise at a rate of d A M dm = c 2 dd dm = c 2 l. Multiplying this figure by the M vehicle miles of driving affected again yields an externality c 2 D. The basic intuition behind the accident externality is simple. If a person decides to go out driving instead of staying at home or using public transportation, she may end up in an accident, and some of the cost of the accident will not be borne by either her or her insurance company; some of the accident cost is borne by the other party to the accident or that party s insurance company (although the average mile is not subsidized, the marginal mile is!) Gains from Per-mile Premiums. We now compare the current insurance system, which we characterize (somewhat unfairly as footnote 3 concedes) as involving lump sum premiums, with two alternative systems: competitive per-mile premiums and Pigouvian per-mile premiums. As derived above, the break-even condition for insurance companies charging per-mile premiums is p = A M = c 1 + c 2 M/l (5) This equation can be viewed as the supply curve for insurance as a function of the number of vehicle miles travelled requiring insurance. Again, in a more sophisticated model, and in practice, rates would vary by risk class i, and break-even competitive prices would be p i = A i M i = c 1i + c 2i M/l, where subscript i s have the natural meaning. Let the utility of each of the n drivers be quasi-linear in the consumption of non-driving goods y and quadratic in miles m: V (y, m) =y + am n b m2. (6) 20 Another way to derive our formula for accidents, in which two-vehicle accidents are proportional to the square of miles driven, is to begin with the premise that the marginal cost of a mile of driving is the expected cost of accidents to both parties that will occur during that mile. Then the marginal cost of accidents will be twice the average: i.e., da 2 car =2 A 2 car. The unique solution to this differential equation, in which dm M the elasticity of accidents with respect to miles is 2, is A 2 car = c 2M 2.

15 12 Then, the aggregate demand will be linear: M = M 0 bp (7) The equilibrium miles, M, and per-mile price, p, are found by solving equations (5) and (7): M = M 0 bc 1 1+bc 2 /l p = c 1 + c 2 M 0 /l. 1+bc 2 /l If drivers continued to drive as much under per-mile premiums as they do under per-year, i.e., if b=0 so that demand were completely inelastic, then insurance companies would break-even by charging p = c 1 + c 2 M 0 /l. however, for b>0, as driving falls in reaction to this charge, the accident rate per-mile will also fall (because there will be fewer cars on the road with whom to collide). As the per-mile accident rate falls, premiums will fall in a competitive insurance industry, as we move down the average cost curve given by Equation (5). Figure 1 depicts the situation. Let c 0 be the non-accident costs of driving (gas, maintenance, etc.) and assume that drivers pay these costs in addition to per-mile insurance premiums p. If drivers pay per year premiums so that p =0, then they demand M 0 miles of driving. The social gain from charging per-mile accident premiums p in this model equals the reduction in accident costs less the lost benefits from foregone driving, the shaded region in Figure 1. This surplus S is given by à S = 1 da + da 2 dm dm M0 M! (M 0 M ) 1 2 p (M 0 M ). (8) The first term is the reduction in accident costs that results from a fall in driving from M 0 to M. The second is the driving benefits lost from this reduction net of the non-accident cost savings c 0 (M 0 M ).

16 13 The marginal accident cost dm da is given by equation (2). Note that because the marginal da accident cost dm lies above the average cost M A, the competitive per-mile premium p is less than the socially optimal accident charge which would lead to M miles driven. Socially optimal accident charges will not result from competition because of the accident externality. µ da dm Government would need to impose a Pigouvian tax of M A 1 100% on insurance M premiums. We call this sum Pigouvian per-mile premiums. By assuming quasi-linear A M utility, we are ignoring income effects. As a per-year premium is shifted to a per-mile charge, under other utility specifications, driving would not fall by as much as it would under a pure price change, because people would no longer have to pay a yearly premium and could use some of that money to purchase more driving than they would under a pure price change. The liklihood of such income effects are, however, overshadowed by our uncertainty about the price responsiveness of driving, so it does not seem worthwhile to consider them explicitly. We ultimately run policy simulations with elasticities of demand chosen conservatively (i.e., on the low side), so our estimates are similar to what they wouldbeifwehadadifferent utility assumption that allowed income effects, but chose a less conservative elasticity. Our benefit calculation assumes that the number of drivers would remain unchanged in a switch to per-mile premiums. In fact, the number of drivers would probably increase under a per-mile system because the total price of a small amount of driving (say 2,000 miles per year) would fall. Although the extra drivers, who drive relatively little, will limit driving reductions and hence accident reductions somewhat, they would probably increase the accident savings net of lost driving benefits, and would surely do so in the case of Pigouvian per-mile premiums. The reason is that these extra drivers gain substantial driving benefits, as evidenced by their willingness to pay insurance premiums. In the case of Pigouvian per-mile premiums, the entry of these extra drivers necessarily increases the benefits from accident cost reductions net of lost driving benefits.

17 14 Pigouvian per-mile premiums could be implemented with a uniform percentage tax on competitive per-mile premiums in either a fault-based tort system or a no-fault tort system, as long as every driver stands an equal chance of being at fault. If drivers differ in fault propensity, then taxing premiums will work better in a no-fault system than in a tort system, because the optimal tax will be invariant to a driver s ability (i.e., invariant to expected share of total damages from relative negligence). To the extent that a no-fault sytem limits recovery to economic damages, as it commonly does in practice, the tax would need to be raised to account for the full externality. 1.2 Congestion Congestion will fall if driving is reduced. In a fundamental respect, congestion is the counterpart to accidents. In the simplest model of congestion, congestion occurs when driver i and j would be in the same location at the same time except that one or both breaks to avoid an accident. The resulting delay is, of course, costly. A rudimentary model of congestion would therefore have congestion costs rising with the square of miles, holding lane miles fixed, so that congestion cost C = a M 2. l As with accident costs, then, the average cost of congestion per-mile would equal one-half the marginal cost: C M = am = 1 µ dc. (9) l 2 dm Equation (9) relates the average cost of delay to the marginal cost, so that we can use Schrank, Turner and Lomax s [1995] estimates of the average cost of delay in order to estimate the marginal cost of delay, and in particular the external marginal cost of delay. This formulation undoubtedly understates the marginal cost (and hence the external cost) of congestion substantially, because as two vehicles slow down they generally force

18 15 others to slow down as well. A cascade of such effects becomes a traffic jam. Looking at measured flow rates of traffic as a function of the number of cars travelling suggests that during periods of congestion the marginal congestion cost of driving is often many times, up to and exceeding 10 times, the average congestion experienced at least during highly congested periods. 21 To be conservative, however, we assume that the marginal cost of congestion is twice the average cost, so that the portion of the marginal cost that is external to the driving decision equals the average cost. Congestion cost savings that are external to the driving decision should also be added to the benefits from per-mile premiums. Assuming, that the mile foregone is a representative mile and not a mile drawn from a particularly congested or uncongested time, the person C foregoing the mile will escape the average cost of delay, M. This savings should not be counted though among our benefits from driving reductions, because it is internalized. Viewed differently, each person derives no net benefit from her marginal mile of driving, because she chooses to drive more miles until driving benefits net of congestion cost just equal operating costs. Yet, as there is less traffic on the road, other drivers will experience reduced delays and this external effect should be added to our calculations. The external effect, as with accidents, equals the difference between the marginal and average cost of delay, so a conservative estimate of the external cost of dm extra miles driven is 2 Data. C M dm. As a proxy for auto accident costs, we use state-level data on total private passenger auto insurance premiums from the National Association of Auto Insurance Commissioners (1998, Table 7). We subtract premiums paid for comprehensive coverage, so that we are left only 21 Author s calculation based upon traffic flow tables. GAO, Traffic Congestion: Trends, Measures, and Effects GAO/PEMD-90-1, November 1989, p. 39.

19 16 with accident coverage. If the insurance industry is competitive, these figures represent the true economic measure of insured accident costs, which includes the administrative cost of the insurance industry and an ordinary return on the capital of that industry. These premium data are for private passenger vehicles, so we adjust these figures to account for commercial premiums by multiplying by 1.14, the national ratio of total premiums to noncommercial premiums. 22 Insured accident costs do not come close to comprising all accident costs. The pain and suffering of at fault drivers is not insured, and auto insurance frequently does not cover their lost wages. (In no-fault states, pain and suffering is also not compensated below certain thresholds). These omitted damages are substantial and their inclusion would raise our estimates of the cost of driving and the benefit of driving reduction significantly. Pain and suffering is often taken to be three times the economic losses from bodily injury. Other data come from a variety of sources. Data on the miles of lanes by state come from Table HM-60, 1996 Highway Statistics, FHWA. Annual vehicle miles by state come from Table VM-2, 1996 Highway Statistics, FHWA. Data on the distribution of fuel efficiency among vehicles in the current U.S. fleet, and the distribution of miles by fuel efficiency of car come from the 1994 Residential Transportation Energy Consumption Survey. We get gasoline prices by state from the Petroleum Marketing Monthly, EIA, Table 31 ( all grades, sales to end users through retail outlets excluding taxes ) and Table EN-1 (federal and state motor gasoline taxes). 3 The traffic density-accident relationship. The social elasticity of accidents with respect to miles of driving should substantially exceed an individual s elasticity because of the externality effect explained in the previous section. Even if the typical individual has an elasticity of.5, the elasticity of total accident costs 22 See, p.22, the Insurance Information Institute 1998 Fact Book.

20 17 with respect to total miles driven would be close to 1 because any individual driver will cause others to have extra accidents when he drives more. One piece of evidence on the social elasticity comes from a study of California freeways from (Lundy, 1964 cited in Vickrey, 1968). A group of 32 segments of four lane freeways with low average traffic had a per-mile accident rate of 1.18 per million miles compared with 1.45 per million miles on twenty segments with more traffic. The implied incremental accident rate was 1.98 accidents per million vehicle miles, suggesting an elasticity of accidents with respect to miles of 1.7=1.98/1.18. Because of the externality associated with driving pointed out in Section 1, we expect the elasticity of total accidents with respect to total miles to exceed the elasticity of an individual s accidents with respect to her driving. In fact, if an individual has an elasticity of 1 as the model assumes, the aggregate elasticity would be 2 if all accidents involved 2 cars. The California highway data accords roughly with what one would predict given that roughly 30% of accidents involve only one vehicle. 23 It is worth comparing accident costs in pairs of states that have similar numbers of lane miles but very different numbers of vehicle miles traveled. For example, New Jersey and Wyoming both have approximately 75,000 lane miles. New Jersey has eight and a half times as much driving, however, and has an average insured accident cost of 7.7 cents per mile traveled instead of the 1.8 cents per mile of Wyoming. Comparing Ohio and Oklahoma we see a similar pattern. Ohio has approximatelytwoandahalftimesasmuchdrivingona similar number of lane miles and has higher average accident cost (3.6 vs. 2.6 cents per mile). Likewise, if we compare Hawaii and Delaware, which have similar numbers of vehicle miles traveled, we find that Hawaii, which has fewer lane miles and so substantially higher traffic density, has substantially higher accident costs per-mile. In general, average accident costs are much higher in states that have a lot more driving, holding lane miles fixed. This feature, which drives the high insurance rates in dense areas, is just another view of the 23 See table 27, U.S. Department of Transportation [1997].

21 18 externality effect. The fact that marginal accident costs are higher than average accident costs is what drives up average accident costs as miles increase. Many other idiosyncratic factors are involved, however, in a state s insurance costs. Maryland and Massachusetts, for example, have an almost identical number of lane miles and fairly similar vehicle miles traveled. However, although Massachusetts drivers only drive about 7 percent more miles per year in aggregate than Maryland drivers their average costs per-mile is 40 percent higher (6.7 cents vs. 4.8 cents), so that total insured accident costs are 45 percent higher. Whether this difference is attributable to differences between Massachusetts and Maryland drivers or differences between the roads or weather in the states is unknown. Cars may also be more expensive to repair in Massachusetts. Here, we fit the model presented in Section 1 in order to form estimates of the marginal accident cost from driving an extra mile in each of the 50 states. As explained in Section 3, we use total auto accident insurance premiums paid in a state as a proxy for the total cost of automobile accidents. We estimate the effects of traffic density on accidents in two ways by a calibration method and a regression method as described below. The regression method utilizes the cross-state variation in traffic density to estimate its effect, while the calibration method relies upon the structure of the model and data on the percentage of accidents involving multiple vehicles. Each method has weaknesses, and after discussing the likely biases in each of these methods, we conclude that the true effect of density lies somewhere between the two estimates. The traffic density effect allows us to estimate the social marginal accident cost of driving and the extent to which this cost exceeds the average, or internalized marginal, cost of driving. We modify the model of Section 1, assuming that each state s idiosyncratic errors ε s enter multiplicatively as follows:

22 19 A s M s = (c 1 + c 2 D s )(1 + ε s ). (10) = c 1s + c 2s D s, (11) where c 1s = c 1 (1 + ε s ) c 2s = c 2 (1 + ε s ) and where s indexes states. Once c 1 and c 2 areestimated,wecanfind the idiosyncratic component ε s for each state from the above equation using the observed values of accident costs, miles traveled and lane miles in the state. We estimate the coefficients c 1 and c 2 in two ways a calibration model and a regression model. In our calibration model, we utilize national data on the percentage of accidents involving multiple cars. Assume that national accident costs are given by A = c 1 M + c 2 MD, where the costs of one- and two-car accidents are, respectively, A 1 = c 1 M and A 2 = c 2 MD. Let a be the average damage per insured vehicle from an accident, so that two-vehicle accidents have total damages of 2a and one-vehicle accidents have damages a. Let r denote

23 20 the proportion of accidents that involve two vehicles. (Nationally, 71% of crashes were multiple-vehicle crashes in 1996, and we assume that multi-car accidents involve only two cars, since we don t have data on the number of cars in multi-car accidents and since this assumption makes our benefit estimate conservative.) 24 If N is the total number of accidents in a state we have: A = N (1 r) a +2Nra, so that Na = A 1+r. This implies that the total cost of one-car accidents is A 1 = (1 r) 1+r A, and similarly for two-car accidents A 2 = 2r 1+r A. The one and two-car accident coefficients can then be determined from the formulas: ĉ 1 = A 1 M = (1 r)a 1+r 1 M and ĉ 2 = A 2 M 2 l = 2r 1+r A l M 2, Using the observed national data on accident costs (A), miles traveled (M), and lane miles (l), we estimate that the one-vehicle coefficient ĉ 1 is roughly.007 dollars per-mile, 24 The statistic 71% is found by taking the ratio of the number of multiple vehicle crashes to total crashes in table 27, U.S. Department of Transportation [1997]. This figure understates the number of accidents that involve multiple vehicles because if a single vehicle crashes into a fixed object, for example, that is a single vehicle crash even if the vehicle swerved to avoid another car.

24 21 while ĉ 2 is 1.1 x 10 7 dollars per-mile squared per lane mile. This means that roughly 18% of costs are attributed to one-car accidents. In our regression model, we estimate the coefficients c 1 and c 2 with a cross-sectional regression. Assuming that the idiosyncratic components ε s are i.i.d. mean zero random variables that are independent of D s, OLS estimates ĉ 1 and ĉ 2 are consistent under standard regularity conditions. Table 1 gives the results of the cross-sectional OLS regression in column 2 and the calibration method in column 1. Theestimateoftheone-vehiclecoefficient ĉ 1 suggests a cost of 2.2 cents per-mile. The other coefficient, ĉ 2, is 5.4 x10 6 cents per squared mile per lane mile. The regression model suggests that 55% of costs are attributable to one-car accidents, i.e., to the linear term. In both models, the marginal accident cost is found by differentiating equation (10) which yields: da s dm s = c 1s +2c 2s D s. We find the state-specific coefficients for one and two vehicle accidents as follows: ˆε s = ĉ 1s =ĉ 1 (1 + ˆε s ) ĉ 2s =ĉ 2 (1 + ˆε s ) A s ĉ 1 M s +ĉ 2 M s D s 1. Table 2 gives the marginal accident costs estimated by the calibration and regression methods. Table 2 allows us to compare these costs with the average accident cost per-mile driven, which appears in column 3. The last row models the U.S. as a whole, treating it as a single state. As we see, accounting for the Vickrey externality appears significant regardless of which method we use, in that the marginal cost of accidents significantly exceeds the average cost. The reason is that both estimation methods put significant positive weight on the quadratic term. The elasticity of accidents with respect to miles (i.e. the ratio of

25 22 marginal to average cost) is higher under the calibration model because that model puts more weight on the quadratic term. Below, we discuss several reasons why the regression estimates probably understate the density effect (and hence the marginal cost of driving), and why the calibration estimates may overstate this effect. The calibration method might overstate accident externalities because the theoretical model does not account for accident substitution i.e., the possibility that if one of the drivers in a two-car accident stayed home, another accident might have substituted for the one that happened. 25 (This bias could be offset, though, by the fact that many accidents require the coincidence of more than two cars at the same place at the same time). A second upward bias results because in the calibration method, c 1 and c 2 are held constant, which does does not account for the fact that as driving becomes more dangerous, drivers and states both take precautionary measures. States react to higher accident rates with higher expenditures on safety by widening roads and lengthening freeway on-ramps. Drivers also make financial expenditures, buying air bags or anti-lock brakes, and nonfinancial expenditures, by paying more attention and slowing down to avoid accidents when driving in heavy traffic. All these precautionary measures mitigate the impact of extra traffic density on accidents. At the margin, if precautions are chosen optimally so that the marginal cost of precautions equals their marginal benefit, then the envelope theorem guarantees that the calibration method would still be properly capturing the sum of accident and prevention costs (i.e., we can treat prevention as being fixed). However, to the extent that people take too little precaution at the moment, the calibration results will overstate the accident externalities. Even if precautions are currently optimal, the calibration results will overstate accident externalities for large changes in behavior, because the marginal analysis of the envelope theorem will not be applicable. 25 For example if vehicle A plows into a line of cars stopped at a light, removing vehicle B from the line might not affect the damage.